Odd numbers balance better if something goes wrong.
Suppose you put four equal weights/loads/fasteners at 90 separation on an axis, each carries 25% of the load. One breaks off; a vibrational load is introduced. Two of the fasteners contribute nothing to restraining vibration around the axis between them. The third fastener is subject to vibrational forces inversely proportional to its distance from the axis of rotation; the wheel is vibrating around an axis where only one lug nut provides restraint.
If there are five fasteners, and four remain intact, the vibrational forces around the axis of maximal assymetry are distributed among four fasteners, each bearing load inversely proportional to its distance from the axis of maximal assymetry. Forces are proportional to the (absolute value of the) cosine of the angular displacement from the failed fastener, and those farthest from the failed fastener bear the greatest stress. This works out to about 15% (each) of what the lug nut on a 4-lug system would bear. Not a bad improvement for adding ONE lug.
On a three-lug wheel, the vibrational forces (on the two remaining lugs) are nominally 25% less than the forces on the single opposite lug of a four-lug system, but the real vibrational force might rapidly become immensely greater as one or both remaining lug nuts loosen even slightly. Total shear capacity under torque would be also be reduced.